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Multiple comparisons and sums of dissociated random variables

Published online by Cambridge University Press:  01 July 2016

A. D. Barbour  and
G. K. Eagleson
A. D. Barbour*
Affiliation:
Gonville and Caius College, Cambridge
G. K. Eagleson*
Affiliation:
CSIRO Division of Mathematics and Statistics
*
∗Present address: Institut für Angewandte Mathematik, Universität Zürich, Rämistrasse 74, 8001 Zürich, Switzerland.
∗∗ Postal address: CSIRO Division of Mathematics and Statistics, P.O. Box 218, Lindfield, NSW 2070, Australia.
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Abstract

Sufficient conditions for a sum of dissociated random variables to be approximately normally distributed are derived. These results generalize the central limit theorem for U-statistics and provide conditions which can be verified in a number of applications. The method of proof is that due to Stein (1970).

Keywords

CENTRAL LIMIT THEOREMRANDOM GRAPHS
Type
Research Article
Information
Advances in Applied Probability , Volume 17 , Issue 1 , March 1985 , pp. 147 - 162
Copyright
Copyright © Applied Probability Trust 1985 

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References

Abe, O. (1969) A central limit theorem for the number of edges in the random intersection of two graphs. Ann. Math. Statist. 40, 144–151.CrossRefGoogle Scholar
Barbour, A. D. and Eagleson, G. K. (1984) Poisson convergence for dissociated statistics. J. R. Statist. Soc. B 46.CrossRefGoogle Scholar
Billingsley, P. (1968) Convergence of Probability Measures. Wiley, New York.Google Scholar
Chen, L. H. Y. (1978) Two central limit problems for dependent random variables. Z. Wahrscheinlichkeitsth. 43, 223–243.CrossRefGoogle Scholar
Cliff, A. D. and Ord, J. K. (1973) Spatial Autocorrelation. Pion, London.Google Scholar
Hoeffding, W. (1948a) A class of statistics with asymptotically normal distribution. Ann. Math. Statist. 19, 293–325.CrossRefGoogle Scholar
Hoeffding, W. (1948b) A nonparametric test of independence. Ann. Math. Statist. 19, 546–557.CrossRefGoogle Scholar
Jonckheere, A. R. (1954) A distribution-free k-sample test against ordered alternatives. Biometrika 41, 133–145.CrossRefGoogle Scholar
Lomnicki, Z. A. and Zaremba, S. K. (1957) A further instance of the central limit theorem for dependent random variables. Math. Z. 66, 490–494.CrossRefGoogle Scholar
Mcginley, W. G. and Sibson, R. (1975) Dissociated random variables. Math. Proc. Camb. Phil. Soc. 77, 185–188.CrossRefGoogle Scholar
Noether, G. E. (1970) A central limit theorem with non-parametric applications. Ann. Math. Statist. 41, 1753–1755.CrossRefGoogle Scholar
Serfling, R. J. (1980) Approximation Theorems of Mathematical Statistics. Wiley, New York.CrossRefGoogle Scholar
Silverman, B. W. (1976) Limit theorems for dissociated random variables. Adv. Appl. Prob. 8, 806–819.CrossRefGoogle Scholar
Silverman, B. W. (1983) Convergence of a class of empirical distribution functions of dependent random variables. Ann. Prob. 11, 745–751.CrossRefGoogle Scholar
Silverman, B. W. and Brown, T. C. (1978) Short distances, flat triangles and Poisson limits. J. Appl. Prob. 15, 815–825.CrossRefGoogle Scholar
Stein, C. (1970) A bound for the error in the normal approximation to the distribution of a sum of dependent random variables. Proc. 6th Berkeley Symp. Math. Statist. Prob. 2, 583–602.Google Scholar